Solution of the Dirac equation in a one-dimensional box

We solve the one-dimensional Dirac equation for a square well potential of depth larger than twice the particle's rest mass energy. The energy spectrum formula for vector coupling in the Klein energy zone is obtained. It matches precisely the transmission resonance spectrum of the relativistic square potential barrier problem. In the nonrelativistic limit, it reduces to the well-known energy spectrum of a particle in a box. We point out the relevance of these results to 1D graphene systems such as graphene nano-tubes and nano-ribbons.